ListConvolve uses Fourier transforms, and it's very difficult to achieve significantly better performance with the FFT for general sparse inputs and outputs than one can get by doing the full transform. So, while the answer to your question is certainly "yes", producing a solution is certainly more than a matter of a simple programming problem.
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Oleksandr R.Nov 20 '12 at 13:44

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If these polynomials are very sparse you can perhaps do it as a sparse matrix-vector multiply, which of course will give you a sparse result. But the performance achievable in this fashion might not be very impressive.
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Oleksandr R.Nov 20 '12 at 13:46

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